scholarly journals The global existence theorem for quasi-linear wave equations with multiple speeds

2004 ◽  
Vol 33 (3) ◽  
pp. 607-636 ◽  
Author(s):  
Kunio HIDANO
Keyword(s):  
Global Existence ◽  
Existence Theorem ◽  
Wave Equations ◽  
Linear Wave ◽  
Global Existence Theorem ◽  
Linear Wave Equations

scholarly journals Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

2018 ◽  
Vol 2020 (1) ◽  
pp. 39-70
Author(s):  
Kunio Hidano ◽  
Kazuyoshi Yokoyama
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Linear Wave ◽  
Null Condition ◽  
Standard Argument ◽  
The Cauchy Problem ◽  
Weight Method ◽  
Linear Wave Equations ◽  
Three Space ◽  
Original Theorem

Abstract We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

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